The problem was to find a single formula which will work to find the area of a parallelogram, a rectangle, a square, a trapezoid and a triangle and explain how it works.

I found the formula by much trial and error with various shapes. Essentially, the problem that I faced was how to deal with a four sided figure as well as a three sided figure, so I felt that the formula had to take into account the number of sides the figure I was dealing with had.

Here is the formula that I ended up with:

A = Area of Figure
h = Altitude of Figure
n = Number of sides
w₁ = Length of one side perpendicular to the altitude.
w₂ = Length of the other side perpendicular to the altitude.

I then went about checking to see if this works for each general figure. I first came up with the general formula for the quadrilaterals and then for the triangles based on n.

Quadrilaterals:

For all of these n = 4 so the equation is reduced to:

Square

In a square w₁ = w₂ = s, where s = the length of one side of the triangle. Substituting these in gives us :

Which is the formula for the area of a square.

Rectangle, parallelogram

In a rectangle and in a parallelogram, w₁ = w₂ = b. Substituting these in gives us:

Which is the formula for the area of a rectangle and a parallelogram.

Trapezoid

In a trapezoid I let w₁ = b₁ which must be the longer of the two bases and w₂ = b₂ which I said was the shorter (although I noticed later that it doesn't matter). Substituting these gave me:

Which is the formula for the area of a trapezoid.

Triangle

This is the one that gave everybody fits. I finally figured that you had to have w₂ = 0 in this case since there is only one base perpendicular to the altitude. This would make w₁ = b. Also different in this case is that n = 3. Substituting all of these into my equation gave me:

Which is the equation for the area of a triangle.

This Solution was developed by Matt Rich for Summer School at North Allegheny Senior High School, Pittsburgh.